Week 2 - Vector Spaces - II - Nptel

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Week 2 - Vector Spaces - II

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Week 0 - Background and Prerequisites

Week 1 - Introduction to Signal Processing, State Space Representation and Vector Spaces - I

Week 2 - Vector Spaces - II

Linear independence and spanning set

Unique representation theorem Basis and cardinality of basis Norms and inner product spaces Inner products and induced norm Cauchy Schwartz inequality Orthonormality

Due on 2019-02-13, 23:59 IST.

1)

1 point

2) 1.5 points

Assignment 02

The due date for submitting this assignment has passed.

As per our records you have not submitted this assignment.

Instructions:

Attempt all questions.

1.

Submission deadline: 13th February 2019 23:55 IST 2.

Solutions to be posted: 14th February 2019 3.

Older browsers might show unnecessary vertical bars at the end of math equations 4.

Consider the vectors and in . Let us consider the inner product,

where . and represent the element of and respectively.

For what value of are the vectors and orthogonal?

No, the answer is incorrect.

Score: 0

Accepted Answers:

(Type: Numeric) 2

Let be a subset of where

Let be the linear span of . What is the dimension of ? 2

3

u = ⎡

⎣ ⎢ 1 1 1

⎤

⎦ ⎥ v = ⎡

⎣ ⎢ 1

−2 3

⎤

⎦ ⎥ R

³

⟨u, v⟩ = u

1

v

1

+ b( u

2

v

2

) + u

3

v

3

b ∈ R u

i

v

i

^th

u v

b u v

S = {a, b, c, d} R

⁵

a = , b = , c = , d =

⎡

⎣

⎢⎢ ⎢⎢

⎢⎢

1 0 1 3 0

⎤

⎦

⎥⎥ ⎥⎥

⎥⎥

⎡

⎣

⎢⎢ ⎢⎢

⎢⎢

0 1 3 1 0

⎤

⎦

⎥⎥ ⎥⎥

⎥⎥

⎡

⎣

⎢⎢ ⎢⎢

⎢⎢

1 1 4 4 0

⎤

⎦

⎥⎥ ⎥⎥

⎥⎥

⎡

⎣

⎢⎢ ⎢⎢

⎢⎢

0 0 0 0 1

⎤

⎦

⎥⎥ ⎥⎥

⎥⎥

L[S] S L[S]

A project of In association with

Funded by

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1 of 4 Wednesday 22 May 2019 10:23 AM

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Week 3 - Vector Spaces - III and Signal Geometry

Week 4 - Probability and Random Processes

Week 5 - Sampling Theorem and Multirate Systems - I

Week 6 - Multirate Systems - II

Week 7 - Multirate Systems - III

Week 8 - Multirate Systems - IV

Week 9 - Wavelets - I

Week 10 - Wavelets - II and Continuity of Functions

Week 11 - Fourier Series - I

Week 12 - Fourier Series - II and KL Transform

Interaction Session

Problem on sum of subspaces Quiz : Assignment 02 Assignment 2 - Solutions

3) 1.5 points

4) 1.5 points

5) 2 points

4 5

No, the answer is incorrect.

Score: 0

3

Consider the vectors and in . Using which of

the following sets can these vectors be represented uniquely?

None of the above No, the answer is incorrect.

Score: 0

The -norm of over is

Score: 0

Consider vectors such that and . Let be the

angle between and . Let be the angle between and . Which of the following relations does and always satisfy?

No relation exists.

u = [1 − 2 3]

^T

v = [−2 4 − 6]

^T

R

³

A = {[1 − 2 3 } ]

^T

B = {[1 0 0 , [0 0 1 } ]

^T

]

^T

C = {[1 − 2 3 , [−2 4 − 6 } ]

^T

]

^T

A = {[1 − 2 3 } ]

^T

L

2

g(x) = 5 − 3x x

²

− 1 [0, 1]

56

√ −−

256

√ −−

76

√ −−

56

√

76

√ −−

x, y, z ∈ R

²

y = x − z ||x|| = ||z|| θ

1

x y θ

2

z y

θ

1

θ

2

|cos θ

1

| = |cos θ

2

|

| cos θ

1

|

²

+ | cos θ

2

|

²

= 1

| cos θ

1

|

²

+ | sin θ

2

|

²

= 1

Mathematical Methods and Techniques in Signa... https://onlinecourses-archive.nptel.ac.in/noc19_...

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6) 2 points

7)

2 points 8)

2.5 points

9) 3 points

Score: 0

If and are orthogonal vectors in the inner product space , such

that and , then what is the value of and ?

Score: 0

Find the dimension of the plane in .

No, the answer is incorrect.

Score: 0

(Type: Numeric) 2

Let and be subspaces of the vector space of matrices defined over integers. Then, subspace is said to be the direct sum

of and if and .The basis of the subspaces are given by

and . For what value

of is not a direct sum?

No, the answer is incorrect.

Score: 0

(Type: Numeric) 3

Let be a linearly independent set in , which of the following are true?

The set is linearly independent.

|cos θ

1

| = |cos θ

2

|

| cos θ

1

|

²

+ | sin θ

2

|

²

= 1

u v V

||u|| = 6 ||u + v|| = 10 ||v|| ||u − v||

4, 2 8, 10 6, 12 8, 28 √ −−

8, 10

2x − 3y = 0 R

³

A B 2 × 2

C

A B C = A + B A ∩ B = {[ 0 ]}

0 0 0

= {[ ] , [ ]}

B

A

m − 12

0 9

m 0 1 m

2 B

B

= {[ m ]}

1 0 m C = A + B 1

S = { , , } u

1

u

2

u

3

R

ⁿ

{ + , + , + } u

1

u

2

u

2

u

3

u

3

u

1

{ − , − , − } u

1

u

2

u

2

u

3

u

3

u

1

Mathematical Methods and Techniques in Signa... https://onlinecourses-archive.nptel.ac.in/noc19_...

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10) 3 points

The set is linearly independent.

No, the answer is incorrect.

Score: 0

The set is linearly independent.

What is the maximum value of subject to the

constraint ?

0 1 2 3

Score: 0

2

{ , + , + + } u

1

u

1

u

2

u

1

u

2

u

3

{ − 2 , − 2 , − 2 } u

1

u

2

u

2

u

3

u

3

u

1

{ + , + , + } u

1

u

2

u

2

u

3

u

3

u

1

{ , + , + + } u

1

u

1

u

2

u

1

u

2

u

3

{ − 2 , − 2 , − 2 } u

1

u

2

u

2

u

3

u

3

u

1

x − 2y + z + 2 + = 1

x

²

y

²

z

²

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